5 research outputs found
Complete directed minors and chromatic number
The dichromatic number χ→(D) of a digraph D is the smallest k for which it admits a k-coloring where every color class induces an acyclic subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several groups of authors have recently studied the containment of complete directed minors in digraphs with a given dichromatic number. In this note we exhibit a relation of these problems to Hadwiger's conjecture. Exploiting this relation, we show that every directed graph excluding the complete digraph K↔t of order t as a strong minor or as a butterfly minor is O(t(log log t)6)-colorable. This answers a question by Axenovich, Girão, Snyder, and Weber, who proved an upper bound of t4t for the same problem. A further consequence of our results is that every digraph of dichromatic number 22n contains a subdivision of every n-vertex subcubic digraph, which makes progress on a set of problems raised by Aboulker, Cohen, Havet, Lochet, Moura, and Thomassé
Adapting the Directed Grid Theorem into an FPT Algorithm
The Grid Theorem of Robertson and Seymour [JCTB, 1986], is one of the most
important tools in the field of structural graph theory, finding numerous
applications in the design of algorithms for undirected graphs. An analogous
version of the Grid Theorem in digraphs was conjectured by Johnson et al.
[JCTB, 2001], and proved by Kawarabayashi and Kreutzer [STOC, 2015]. Namely,
they showed that there is a function such that every digraph of directed
tree-width at least contains a cylindrical grid of size as a
butterfly minor and stated that their proof can be turned into an XP algorithm,
with parameter , that either constructs a decomposition of the appropriate
width, or finds the claimed large cylindrical grid as a butterfly minor. In
this paper, we adapt some of the steps of the proof of Kawarabayashi and
Kreutzer to improve this XP algorithm into an FPT algorithm. Towards this, our
main technical contributions are two FPT algorithms with parameter . The
first one either produces an arboreal decomposition of width or finds a
haven of order in a digraph , improving on the original result for
arboreal decompositions by Johnson et al. The second algorithm finds a
well-linked set of order in a digraph of large directed tree-width. As
tools to prove these results, we show how to solve a generalized version of the
problem of finding balanced separators for a given set of vertices in FPT
time with parameter , a result that we consider to be of its own interest.Comment: 31 pages, 9 figure
A more accurate view of the Flat Wall Theorem
We introduce a supporting combinatorial framework for the Flat Wall Theorem.
In particular, we suggest two variants of the theorem and we introduce a new,
more versatile, concept of wall homogeneity as well as the notion of regularity
in flat walls. All proposed concepts and results aim at facilitating the use of
the irrelevant vertex technique in future algorithmic applications.Comment: arXiv admin note: text overlap with arXiv:2004.1269